Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge

TL;DR

This paper investigates fixed energy solutions to Euler-Lagrange equations with indefinite Lagrangians, utilizing symmetries and Noether charges to reduce the problem to a boundary value problem, extending Fermat's principle in certain cases.

Contribution

It introduces a novel approach to fixed energy solutions using symmetry and Noether charge invariance, extending classical variational principles to indefinite Lagrangians.

Findings

Derived an equation involving the differential of the arrival time functional.

Extended Fermat's principle to stationary spacetimes with indefinite Lagrangians.

Analyzed solutions with affine Noether charge and their properties.

Abstract

We consider an autonomous, indefinite Lagrangian admitting an infinitesimal symmetry whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point to a flow line of the infinitesimal generator $K$. By utilizing the invariance of the Lagrangian under the flow of $K$, we simplify the problem into a two-point boundary problem. Consequently, we derive an equation that involves the differential of the ``arrival time'', seen as a functional on the infinite dimensional manifold of connecting paths satisfying the semi-holonomic constraint defined by the Noether charge. When the Lagrangian is positively homogeneous of degree two in the velocities, the resulting equation establishes a variational principle that extends the Fermat's principle in a stationary spacetime. Furthermore, we also analyze the scenario where the Noether charge is affine.